program your own particles
I don't believe you can use python to add a new particle simulation type. While you can use python to access the particle system data such as particle location, I don't think you will have any luck altering the particle positions.

Before you start coding yourself I would suggest looking at sverchok and animation nodes. There is an introduction to animation nodes here and you will find blendersushi has many videos of experimenting with both addons. If they can't accomplish what you want to do, they may be a starting point for your own project or maybe you could contribute your own custom node to one of them.